What a great itinerary! Lucky you! Gramercy tavern, wow, haven't been there in forever--I'm jealous!
I love LA, hard to know where to suggest staying as it is indeed quite spread out. I like Beverly hills near LACMA, rodeo drive, the tar pits--it's pretty walkable and there are buses on the main drag.Or if you stay in Venice/Santa Monica, plenty to do near there within walking distance.
Cheap NYC things abound:
Ride the staten island ferry to see the city skyline and the statue of liberty (deboard and reboard at the SI terminal, it's very easy)
Visit the Met museum or AMNH, both donation based--wander in Central Park before or after, or walk from one to the other through the park if the weather is good! October is a wonderful time of year to visit
The Cloisters is a great museum too, way uptown in Manhattan (Met affiliated, also donation based)
You might still be in season to go to coney island if that's your thing and the weather is warm (the wonder wheel should be operational on weekends until the end of October), you could also ride the tram to roosevelt island for the cost of a metro ticket (2.50)
Eat in chinatown, murray hill ("curry hill")
See free shows at the UCB (usually these are late night) or elsewhere
I'd skip the NY pass, but I live here so I have more leisure time to see everything.
I also love the views from the brooklyn heights promenade, and brooklyn bridge park is great. Walk from downtown to dumbo on the brooklyn bridge, or vice versa. You can get a pizza at grimaldis in dumbo, at strange hours of the afternoon it is not that packed.
There is hardly any need to spend money on admission to anything because there are so many neighborhoods to wander, cheap foods to eat, and free cultural resources.
Definitely check out the full length of the high line in chelsea--I believe the new whitney will be pay what you wish on friday nights, too. The high line can be crowded, so off hours are better (assuming late at night or early in the morning)
Times square is an obligatory stop but terrible. If you like theater however you can get rush or lottery tickets at some places. I think that Book of Mormon has a day of lottery and is supposed to be worth it. I would not book anything in advance besides Alcatraz, event wise.
Also, bring comfortable shoes, be prepared to walk, and be prepared for a range of weather options.